BOOK REVIEW

Nominalism and the application of mathematics

Jody Azzouni: Deflating existential consequence:A case for nominalism. New York: Oxford University Press,2004. viii+241pp, $49.95 HB

Otavio Bueno

Published online: 20 March 2012

� Springer Science+Business Media B.V. 2012

A significant feature of contemporary science is the widespread use of mathematics

in several of its subfields. In many instances, the content of scientific theories cannot

be formulated without reference to mathematical objects (such as functions,

numbers, or sets). In the hands of W. V. Quine and Hilary Putnam, this feature of

scientific practice was invoked in support of platonism (the view according to which

mathematical objects exist). Quine and Putnam insisted that one ought to be

ontologically committed to mathematical entities since they are indispensable to our

best theories of the world. This is the indispensability argument.

This argument posed a formidable challenge to nominalists, who now needed to

show either (i) that mathematical entities are not indispensable to mathematics or (ii)

that quantification over these entities does not require ontological commitment. Since

Hartry Field’s Science without Numbers (Princeton, NJ: Princeton University Press,

1980), most nominalisation strategies have attempted to show that mathematics is

ultimately dispensable, thus taking up (i): Geoffrey Hellman’s modal structuralism

and Charles Chihara’s constructibility approach offer two examples. Unfortunately,

for a variety of technical and philosophical reasons, none of these strategies have

succeeded (see John Burgess and Gideon Rosen, A Subject with No Object, Oxford:

Oxford University Press, 1997).

In his book, Jody Azzouni provides the most thorough and detailed attempt to

explore (ii). On his view, (a) mathematical theories are in fact indispensable to

science. It is often not even possible to express the content of a scientific theory

without invoking mathematical objects. Moreover, (b) mathematical and scientific

theories are true, and they need to be taken to be true, since often one needs to draw

consequences from such theories without being able to specify exactly what their

content is. Despite (a) and (b), however, Azzouni denies that mathematical objects

exist. How?

O. Bueno (&)

Department of Philosophy, University of Miami, Coral Gables, FL 33124, USA

e-mail: otaviobueno@me.com

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DOI 10.1007/s11016-012-9653-6

Although quantification over mathematical objects and relations is indispensable

to our best theories of the world, there is no reason to believe in the existence of the

corresponding entities. As Azzouni notes, one needs to distinguish two kinds of

commitment: quantifier commitment and ontological commitment (127; see also

49–122). We incur a quantifier commitment whenever our theories imply existentially

quantified statements. But existential quantification is not sufficient for ontological

commitment. We often quantify over objects we have no reason to believe exist, such

as fictional entities.

To incur an ontological commitment—that is, to be committed to the existence of

a given object—a criterion for what exists needs to be satisfied. There are, of course,

various possible criteria for what exists (such as causal efficacy, observability,

possibility of detection, and so on). But the criterion Azzouni favours, and he takes

it to be the one that has been collectively adopted, is ontological independence (99).

What exist are the things that are ontologically independent of our linguistic

practices and psychological processes. The point is that if we have just made

something up by saying or thinking of something, there is no need for us to be

committed to the existence of the corresponding object. And typically, we would

resist any such commitment.

Quine, of course, identifies quantifier and ontological commitments, at least in

the crucial case of the objects that are indispensable to our best theories of the

world. Such objects are those that cannot be eliminated through paraphrase and over

which we have to quantify when we regiment the relevant theories (using first-order

logic). According to Quine’s criterion, these are precisely the objects we are

ontologically committed to. Azzouni insists that we should resist this identification.

Even if the objects in our best theories are indispensable, even if we quantify over

them, this is not sufficient for us to be ontologically committed to them. The objects

we quantify over might be ontologically dependent on us—they may depend on our

linguistic practices or psychological processes—and thus we might have just made

them up. In this case, clearly there is no reason to be committed to their existence.

However, for those objects that are ontologically independent of us, we arecommitted to their existence.

As it turns out, on Azzouni’s view, mathematical objects are ontologically

dependent on our linguistic practices and psychological processes. And so, even

though they may be indispensable to our best theories of the world, we are not

ontologically committed to them. Hence, as advertised, we have here a form of

nominalism.

But in what sense do mathematical objects depend on our linguistic practices and

psychological processes? In the sense that the sheer postulation of certain principles

is enough for mathematical practice: ‘A mathematical subject with its accompa-

nying posits can be created ex nihilo by simply writing down a set of axioms’ (127).

The only additional constraint that sheer postulation has to meet, in practice, is that

mathematicians should find the resulting mathematics interesting. That is, the

consequences that follow from the relevant mathematical principles should not be

obvious, and they should be computationally tractable. Thus, given that sheer

postulation is (basically) enough in mathematics, mathematical objects have no

epistemic burdens. Such objects, or ‘posits’, are called ultrathin (127).

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The same move that Azzouni makes to distinguish ontological commitment from

quantifier commitment is also used to distinguish ontological commitment to Fs

from asserting the truth of ‘There are Fs’. Although mathematical theories used in

science are (taken to be) true, this is not sufficient to commit us to the existence of

the objects these theories are supposedly about. It may be true that there are Fs, but

to be ontologically committed to Fs, a criterion for what exists needs to be satisfied.

As Azzouni (4–5) points out:

I take true mathematical statements as literally true; I forgo attempts to show

that such literally true mathematical statements are not indispensable to

empirical science, and yet, nonetheless, I can describe mathematical terms as

referring to nothing at all. Without Quine’s criterion to corrupt them,

existential statements are innocent of ontology.

On this picture, ontological commitment is not signalled in any special way in

natural (or even formal) language. We just do not read off the ontological

commitment of scientific doctrines (even if they were suitably regimented). Without

Quine’s criterion of ontological commitment, neither the quantification over objects

(in a first-order language) nor formulation of true claims about them entails their

existence.

But does this move go through? I think the form of nominalism Azzouni favours

is defensible, with some adjustments. But the underlying scientific realism is not

required. Quine, who was a realist in science, was forced by the indispensability

argument to become a grudging platonist about mathematics. Having resisted the

indispensability argument, thus opening up the path to nominalism, Azzouni still

lingers in the scientific realist landscape. Why?

First, as opposed to Azzouni’s move, scientific theories need not be taken to be

true. They can be taken to be empirically adequate only, as empiricists such as Bas

van Fraassen have insisted. Logical consequences can be drawn from such theories

without invoking a truth predicate. In fact, logical consequence is better thought of

as a modal notion, according to which B follows A as long as A and the negation of

B is not possible (possibility, a primitive modal notion, is not here understood in

terms of truth or via mathematical models). The impossibility in question emerges

from the fact that A and the negation of B yield a contradiction when asserted

together. Truth is not needed here—or elsewhere.

Second, Azzouni’s ontological independence criterion backfires: the platonist is

the first to grant that mathematical objects exist independently of our linguistic

practices or psychological processes. Rather than nominalism, the criterion yields a

form of platonism.

A better alternative consists in granting that such metaphysical disputes, about

whether mathematical objects exist or not, probably cannot be answered satisfac-

torily. No wonder these disputes have been with us for so long. It is not clear,

though, how such disputes can be ultimately settled. If we insisted that only

spatiotemporally located objects exist, platonists will rightly complain that we have

begged the question against them. If we insisted that ontological commitment to

mathematical objects is needed in order to express claims about the physical world,

nominalists will rightly complain that this is not the case. In situations such as this,

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the more reasonable approach is just to note that the existence of mathematical

objects does not play any role in mathematical or scientific practice—a point

Azzouni correctly emphasizes. Settling the issue about the existence (or not) of

these objects is not required. As good empiricists, we can remain agnostic about the

issue, and still have plenty of room to make sense of the use of mathematics in

science. If not for Azzouni’s commitment to scientific realism, this should have

been the conclusion he reached at the end of this stimulating and provocative book.

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